On the solution of integral equations with a generalized Cauchy kernel

Abstract

In this paper a certain class of singular integral equations that may arise from the mixed boundary value problems in nonhomogeneous materials is considered. The distinguishing feature of these equations is that in addition to the Cauchy singularity, the kernels contain terms that are singular only at the end points. In the form of the singular integral equations adopted, the density function is a potential or a displacement and consequently the kernel has strong singularities of the form ( t − x ) − 2 , x n − 2 ( t + x ) n ( n ≥ 2 , 0 > x , t > b ) {\left ( {t - x} \right )^{ - 2}},{x^{n - 2}}{\left ( {t + x} \right )^n}\left ( {n \ge 2,0 > x,t > b} \right ) . The complex function theory is used to determine the fundamental function of the problem for the general case and a simple numerical technique is described to solve the integral equation. Two examples from the theory of elasticity are then considered to show the application of the technique.